On the FT backend, `PH' can reduce the CNOT, single-qubit, total gate count, and circuit depth by 38.7% (44.5%), 18.6% (3.0%), 32.8% (34.4%), and 61.7% (68.0%), respectively on average, compared with `TK' using `Qiskit_L3' (`tket_O2').
The circuit depth reduction is significant due to the depth-oriented scheduling in `PH'. Our block-wise optimization is also much effective compared with `TK' strategy.
The details of `TK' are not public and what we can infer, at our best, from their limited documents [1]}, [2]}, [3]}, [4]} is that the simultaneous diagonalization may introduce too much overhead.
For example, the `Ising-1D' program has even more gates after `TK'. One possible reason is that all Pauli strings in Ising-1D are mutually commutative and it takes many additional gates to simultaneously diagonalize all these Pauli strings.
